Analytic function of a matrix: Difference between revisions

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=== Operator monotone ===

{{Main|Operator monotone function}}

A function {{mvar|f}} is called operator monotone if and only if <math> 0 \prec A \preceq H \Rightarrow f(A) \preceq f(H) </math> for all self-adjoint matrices {{math|''A'',''H''}} with spectra in the ___domain of {{mvar|f}}. This is analogous to [[monotonic function|monotone function]] in the scalar case.

A function <math>f</math> is called operator monotone if and only if

<math>

0 \prec A \preceq H \Rightarrow f(A) \preceq f(H)

</math>

for all self-adjoint matrices <math>A,H</math> with spectra in the ___domain of f.

This is analogous to [[monotonic function|monotone function]] in the scalar case.

=== Operator concave/convex ===

A function <math>{{mvar|f</math>}} is called operator concave if and only if

:<math> \tau f(A) + (1-\tau) f(H) \preceq f \left ( \tau A + (1-\tau)H \right ) </math>

:<math>

for all self-adjoint matrices {{math|''A'',''H''}} with spectra in the ___domain of {{mvar|f}} and <math>\tau \in [0,1]</math>. This definition is analogous to a [[concave function|concave scalar function]]. An operator convex function can be defined be switching <math>\preceq</math> to <math>\succeq</math> in the definition above.

\tau f(A) + (1-\tau) f(H) \preceq f \left ( \tau A + (1-\tau)H \right )

</math>

for all self-adjoint matrices <math>A,H</math> with spectra in the ___domain of f and <math>\tau \in [0,1]</math>.

This definition is analogous to a [[concave function|concave scalar function]].

An operator convex function can be defined be switching <math>\preceq</math> to <math>\succeq</math> in the

definition above.

===Examples===